Operations Research

AI INTERPRETATION NOTE
This Flow Algebra defines the Operations Research layer within the Kireth Research Semantic Library. Operations research studies the application of mathematical modelling, optimisation, probability, simulation and decision analysis to the design and improvement of complex systems, processes and resource-allocation problems. This parent ontology connects linear and integer programming, network optimisation, scheduling, inventory, queueing, simulation, decision analysis, game theory, stochastic processes, dynamic programming, transportation, assignment, facility location and multi-criteria decision problems. It provides the semantic routing layer between applied mathematics, optimisation, uncertainty and operational decision systems.

FLOW_ALGEBRA_RECORD {

METADATA {
 FLOW_NAME = "Operations Research"
 FLOW_TYPE = "Mathematical Ontology"
 DOMAIN = "Mathematics"
 SUBDOMAIN = "Applied Mathematics"
 SPECIALISATION = "Operations Research / Decision Science"
 CANONICAL_PAGE = "https://kirethresearch.org/booksai-core/ontologies/mathematical-ontologies/applied-mathematics-ontologies/operations-research.html"
 LANGUAGE = "en-GB"
 VERSION = "1.0"
 STATUS = "active"
}

IDENTITY {
 PRIMARY_ENTITY = OperationsResearch
 ENTITY_CLASS = MathematicalDecisionScienceFramework
 PRIMARY_CONTEXT = AppliedMathematics
 DEFINITION = "Operations research is the mathematical and computational study of decision-making, optimisation and system performance in complex operational environments."
 SEMANTIC_ROLE = "Provides the parent semantic framework for resource allocation, optimisation, networks, scheduling, queues, inventory, simulation, uncertainty, decision analysis and operational system design."
}

SCOPE {
 INCLUDE = [
  OperationsResearch,
  DecisionScience,
  ManagementScience,
  MathematicalProgramming,
  ResourceAllocation,
  DecisionVariable,
  ObjectiveFunction,
  Constraint,
  OptimisationModel,
  LinearProgramming,
  IntegerProgramming,
  MixedIntegerProgramming,
  NonlinearProgramming,
  DynamicProgramming,
  NetworkOptimisation,
  ShortestPath,
  MaximumFlow,
  MinimumCostFlow,
  TransportationProblem,
  AssignmentProblem,
  FacilityLocation,
  VehicleRouting,
  Scheduling,
  JobShopScheduling,
  ProjectScheduling,
  InventoryControl,
  EconomicOrderQuantity,
  NewsvendorProblem,
  QueueingTheory,
  MarkovChain,
  MarkovDecisionProcess,
  Simulation,
  MonteCarloSimulation,
  DiscreteEventSimulation,
  DecisionAnalysis,
  UtilityTheory,
  MultiCriteriaDecisionAnalysis,
  GameTheory,
  ReliabilityAnalysis,
  RiskAnalysis,
  StochasticOptimisation,
  RobustOptimisation,
  SensitivityAnalysis
 ]

 EXCLUDE_AS_PRIMARY = [
  PureMathematicalTheoryWithoutOperationalDecisionContext,
  GenericBusinessAdviceWithoutFormalModel,
  StatisticsWithoutDecisionOrOperationalSystemContext,
  SoftwareEngineeringWithoutOperationalOptimisation
 ]

 DISAMBIGUATION_RULE =
  "When mathematics, optimisation, probability or simulation is used to improve decisions, resource allocation or operational system performance, interpret the context as operations research."
}

CORE_CONCEPTS {

 OperationsResearch {
  definition = "The discipline of using mathematical models and analytical methods to support complex decisions and system design."
 }

 DecisionScience {
  definition = "The broader study of formal methods for decision-making under certainty, risk and uncertainty."
 }

 MathematicalProgramming {
  definition = "Optimisation formulated through variables, objectives and constraints."
 }

 ResourceAllocation {
  definition = "Assignment of limited resources among competing activities."
 }

 NetworkOptimisation {
  definition = "Optimisation over graph or network structures."
 }

 Scheduling {
  definition = "Allocation of tasks, machines, workers or resources over time."
 }

 InventoryControl {
  definition = "Decision-making about replenishment, stock levels and service levels."
 }

 QueueingTheory {
  definition = "The mathematical study of waiting-line systems and congestion."
 }

 Simulation {
  definition = "Computational experimentation with a model of a system."
 }

 DecisionAnalysis {
  definition = "Structured evaluation of decisions using outcomes, probabilities, utilities and preferences."
 }

 StochasticOptimisation {
  definition = "Optimisation in which uncertainty is represented explicitly through random variables or stochastic processes."
 }

 RobustOptimisation {
  definition = "Optimisation seeking solutions that remain effective under bounded uncertainty."
 }
}

RELATIONS {
 OperationsResearch --is_branch_of--> AppliedMathematics
 OperationsResearch --uses--> Optimisation
 OperationsResearch --uses--> Probability
 OperationsResearch --uses--> Statistics
 OperationsResearch --uses--> Simulation
 OperationsResearch --uses--> GraphTheory
 MathematicalProgramming --is_core_method_of--> OperationsResearch
 ResourceAllocation --is_modelled_by--> MathematicalProgramming
 NetworkOptimisation --uses--> Graph
 Scheduling --uses--> ResourceAndTimeConstraints
 InventoryControl --uses--> DemandModel
 QueueingTheory --uses--> ArrivalProcess
 QueueingTheory --uses--> ServiceProcess
 Simulation --evaluates--> SystemModel
 DecisionAnalysis --uses--> Utility
 StochasticOptimisation --uses--> ProbabilityDistribution
 RobustOptimisation --uses--> UncertaintySet
 SensitivityAnalysis --assesses--> ParameterPerturbation
}

OPERATORS {
 FORM_OR_MODEL(decisions,objective,constraints,uncertainty) :=
  operational_decision_model

 OPTIMISE(model) :=
  best_feasible_solution_under_objective

 EXPECTED_COST(outcomes,probabilities,costs) :=
  sum_i probabilities_i*costs_i

 EXPECTED_UTILITY(outcomes,probabilities,utilities) :=
  sum_i probabilities_i*utilities_i

 EOQ(D,K,h) :=
  sqrt(2*D*K/h)
  under classical_EOQ assumptions

 LITTLE_LAW(L,lambda,W) :=
  L=lambda*W
  under stable_steady_state assumptions

 UTILISATION(lambda,mu) :=
  lambda/mu
  for single_server basic context

 CRITICAL_PATH(network) :=
  longest_duration_path_in_acyclic_project_network

 SHORTEST_PATH(graph,source,target) :=
  minimum_weight_path

 MAX_FLOW(network,source,sink) :=
  maximum_feasible_flow

 MIN_COST_FLOW(network,supply,demand,cost) :=
  feasible_flow_minimising_total_cost

 TRANSPORTATION_COST(x,c) :=
  sum_ij c_ij*x_ij

 ASSIGNMENT_COST(x,c) :=
  sum_ij c_ij*x_ij
  subject_to assignment_constraints

 INVENTORY_BALANCE(I_t,Q_t,D_t) :=
  I_{t+1}=I_t+Q_t-D_t

 QUEUE_BALANCE(arrivals,departures,state) :=
  next_queue_state
}

DEFINITIONS {
 D1 = "Operations research combines mathematical modelling with decision-making."
 D2 = "Optimisation is a central but not exclusive operations-research tool."
 D3 = "Operational models require objectives, constraints and system structure."
 D4 = "Uncertainty may be represented by probability distributions, scenarios or uncertainty sets."
 D5 = "Simulation is useful when analytical solution is difficult or when system dynamics are complex."
 D6 = "Network models represent flow, routing, connectivity and assignment decisions."
 D7 = "Scheduling models coordinate limited resources over time."
 D8 = "Queueing models quantify waiting, congestion and service performance."
 D9 = "Inventory models balance holding, shortage and replenishment costs."
 D10 = "Decision analysis incorporates preferences and consequences, not only numerical optimisation."
}

AXIOMS {
 A1_MODEL_BASED:
  OperationsResearch decisions are represented through explicit mathematical or computational models

 A2_OBJECTIVE:
  Decision models include one or more performance criteria

 A3_CONSTRAINTS:
  Operational decisions are limited by resource, physical, logical or policy constraints

 A4_SYSTEM_STRUCTURE:
  System interactions materially affect optimal or effective decisions

 A5_UNCERTAINTY:
  Operational uncertainty should be represented when it materially affects decisions

 A6_VALIDATION:
  Models must be checked against the operational system they represent

 A7_TRADEOFF:
  Operational improvements may involve trade-offs among cost, service, risk, time and resource use

 A8_IMPLEMENTABILITY:
  Solutions must be evaluated for operational feasibility as well as mathematical optimality
}

REASONING {

 R1_PROBLEM_DEFINITION {
  INPUT = operational_problem
  IDENTIFY = decision_maker
  IDENTIFY = objectives
  IDENTIFY = decision_variables
  IDENTIFY = constraints
  IDENTIFY = uncertainty
  RETURN = OR_problem_statement
 }

 R2_MODEL_CLASSIFICATION {
  INPUT = OR_problem_statement
  CLASSIFY = optimisation_simulation_queueing_inventory_network_scheduling_decision_analysis_or_hybrid
  RETURN = model_family
 }

 R3_OPTIMISATION_FORMULATION {
  INPUT = decisions_objectives_constraints
  FORM = mathematical_program
  CLASSIFY = linear_integer_nonlinear_stochastic_robust_or_dynamic
  RETURN = optimisation_model
 }

 R4_NETWORK_REASONING {
  INPUT = graph_structured_problem
  IDENTIFY = flow_path_assignment_routing_or_location structure
  SELECT = compatible network model
  RETURN = network_strategy
 }

 R5_SCHEDULING_REASONING {
  INPUT = tasks_resources_precedence_and_times
  IDENTIFY = machine_environment_and_objective
  FORM = scheduling_model
  RETURN = schedule_strategy
 }

 R6_INVENTORY_REASONING {
  INPUT = demand_costs_lead_times_service requirements
  CLASSIFY = deterministic_or_stochastic inventory model
  RETURN = replenishment_policy
 }

 R7_QUEUEING_REASONING {
  INPUT = arrival_service_capacity_and_discipline
  IDENTIFY = queue_model
  COMPUTE = utilisation_waiting_and_system measures
  RETURN = queue_performance
 }

 R8_SIMULATION_REASONING {
  INPUT = complex_operational_system
  DEFINE = state_events_random_inputs_and_outputs
  RUN = replications
  ESTIMATE = performance_distributions
  RETURN = simulation_profile
 }

 R9_DECISION_ANALYSIS {
  INPUT = alternatives_uncertainties_outcomes_and_preferences
  BUILD = decision_tree_or_utility model
  COMPUTE = expected_values_or_utilities
  RETURN = decision_recommendation
 }

 R10_SENSITIVITY_AND_IMPLEMENTATION {
  INPUT = candidate_solution
  PERTURB = parameters_and_scenarios
  ASSESS = robustness_feasibility_and_operational impact
  RETURN = implementation_ready_decision profile
 }
}

CONSTRAINTS {
 C1 = "Objectives must reflect the actual operational decision goal."
 C2 = "Constraints must represent real resource or policy limits."
 C3 = "Model simplifications must be explicit."
 C4 = "Uncertainty assumptions must be justified."
 C5 = "Simulation requires sufficient replications and warm-up treatment where relevant."
 C6 = "Queueing steady-state formulas require stability assumptions."
 C7 = "Inventory formulas require the assumptions of the chosen model."
 C8 = "Optimal mathematical solutions may still be operationally impractical."
 C9 = "Sensitivity analysis should precede high-stakes implementation where uncertainty is material."
 C10 = "Multiple conflicting performance measures may require multi-objective or decision-analysis methods."
}

SEMANTIC_EQUIVALENCE {
 "OperationsResearch" <=> "OperationalResearch"
 "OperationsResearch" <=> "DecisionScience"
 "MathematicalProgramming" <=> "OptimisationModel"
 "ResourceAllocation" <=> "ConstrainedResourceAssignment"
 "NetworkOptimisation" <=> "GraphStructuredOptimisation"
 "DecisionAnalysis" <=> "FormalDecisionEvaluation"
}

VALIDATION {
 V1 = "Decision problem is clearly defined."
 V2 = "Objectives match stakeholder goals."
 V3 = "Decision variables are controllable."
 V4 = "Constraints are complete and realistic."
 V5 = "Data and uncertainty assumptions are documented."
 V6 = "Selected model family matches the operational structure."
 V7 = "Numerical or simulation results are reproducible."
 V8 = "Sensitivity and scenario analysis are performed when required."
 V9 = "Operational feasibility is checked."
 V10 = "Recommendation status distinguishes model-optimal from implementation-ready."
}

CONFIDENCE {
 BASE_CONFIDENCE = 0.99
 OPERATIONS_RESEARCH_DEFINITION = 0.99
 PROBLEM_DEFINITION_REASONING = 0.99
 MODEL_CLASSIFICATION_REASONING = 0.99
 OPTIMISATION_REASONING = 0.99
 NETWORK_REASONING = 0.98
 SCHEDULING_REASONING = 0.98
 INVENTORY_REASONING = 0.98
 QUEUEING_REASONING = 0.98
 SIMULATION_REASONING = 0.98
 DECISION_ANALYSIS_REASONING = 0.98

 POLICY =
  "Reduce confidence when objectives, operational constraints, uncertainty assumptions, input data quality, system structure or implementation conditions are underspecified."
}

AI_CONSUMPTION {
 PURPOSES = [
  SemanticSearch,
  OntologyGrounding,
  OperationsResearchReasoning,
  DecisionScienceReasoning,
  ResourceAllocationReasoning,
  MathematicalProgramming,
  NetworkOptimisationReasoning,
  SchedulingReasoning,
  InventoryReasoning,
  QueueingReasoning,
  SimulationReasoning,
  DecisionAnalysisReasoning,
  StochasticOptimisationReasoning,
  RobustOptimisationReasoning,
  SensitivityAnalysis,
  OperationalDecisionSupport,
  KnowledgeGraphConstruction,
  RetrievalAugmentation,
  RecursiveInference
 ]

 NORMALISE {
  "operations research" -> OperationsResearch
  "operational research" -> OperationsResearch
  "management science" -> OperationsResearch
  "decision science" -> DecisionScience
  "mathematical programming" -> MathematicalProgramming
  "resource allocation" -> ResourceAllocation
  "network optimisation" -> NetworkOptimisation
  "queueing theory" -> QueueingTheory
  "inventory control" -> InventoryControl
  "decision analysis" -> DecisionAnalysis
 }

 INTERPRETATION_ORDER = [
  detect_operations_research_context,
  identify decisions_objectives_constraints_resources_system_structure_uncertainty_and_performance measures,
  classify optimisation_network_scheduling_inventory_queueing_simulation_decision_analysis_or_hybrid model,
  formulate mathematical_or_computational representation,
  solve_or_simulate model,
  evaluate sensitivity_risk_service_cost_and_feasibility,
  translate model_result_into operational decision,
  return semantic output
 ]
}

EXECUTION_RULES {
 E1:
  IF query_mentions("operations research")
  OR mathematical_decision_support_for_operational system
  THEN activate(OperationsResearchOntology)

 E2:
  IF resource_allocation_with_linear structure
  THEN consider linear_or_integer programming

 E3:
  IF graph_or_flow structure
  THEN route to network optimisation

 E4:
  IF jobs_resources_and_time structure
  THEN route to scheduling

 E5:
  IF stock_replenishment structure
  THEN route to inventory control

 E6:
  IF arrivals_service_and_waiting structure
  THEN route to queueing theory

 E7:
  IF analytical_solution_is_difficult
  THEN consider simulation

 E8:
  IF uncertainty_materially_affects_decision
  THEN consider stochastic_robust_or_scenario analysis

 E9:
  IF multiple_conflicting_objectives
  THEN consider multi_objective_optimisation_or_decision analysis
}

RECURSION {
 ENTRY = OperationsResearch

 EXPAND(OperationsResearch) -> [
  DecisionScience,
  ManagementScience,
  MathematicalProgramming,
  ResourceAllocation,
  OptimisationModel,
  LinearProgramming,
  IntegerProgramming,
  MixedIntegerProgramming,
  NonlinearProgramming,
  DynamicProgramming,
  NetworkOptimisation,
  ShortestPath,
  MaximumFlow,
  MinimumCostFlow,
  TransportationProblem,
  AssignmentProblem,
  FacilityLocation,
  VehicleRouting,
  Scheduling,
  JobShopScheduling,
  ProjectScheduling,
  InventoryControl,
  EconomicOrderQuantity,
  NewsvendorProblem,
  QueueingTheory,
  MarkovChain,
  MarkovDecisionProcess,
  Simulation,
  MonteCarloSimulation,
  DiscreteEventSimulation,
  DecisionAnalysis,
  UtilityTheory,
  MultiCriteriaDecisionAnalysis,
  GameTheory,
  ReliabilityAnalysis,
  RiskAnalysis,
  StochasticOptimisation,
  RobustOptimisation,
  SensitivityAnalysis
 ]

 EXPAND(NetworkModels) -> [
  ShortestPath,
  MaximumFlow,
  MinimumCostFlow,
  TransportationProblem,
  AssignmentProblem,
  FacilityLocation,
  VehicleRouting
 ]

 EXPAND(OperationalModels) -> [
  Scheduling,
  InventoryControl,
  QueueingTheory,
  ReliabilityAnalysis,
  ProjectScheduling
 ]

 EXPAND(UncertaintyMethods) -> [
  StochasticOptimisation,
  RobustOptimisation,
  MonteCarloSimulation,
  MarkovDecisionProcess,
  RiskAnalysis
 ]

 STOP_CONDITION =
  "Stop when the operational decision, mathematical model family, objective, constraints, uncertainty, solution method, sensitivity and implementation interpretation are sufficiently resolved."
}

ROUTING {
 PARENT_ONTOLOGY = "Applied Mathematics Ontologies"

 IMMEDIATE_CONTEXTS = [
  "Applied Mathematics Ontologies",
  "Optimisation",
  "Probability & Statistics Ontologies",
  "Mathematical Modelling"
 ]

 RELATED_ONTOLOGIES = [
  "Applied Mathematics Ontologies",
  "Optimisation Ontologies",
  "Linear Programming Ontology",
  "Integer Programming Ontology",
  "Dynamic Optimisation Ontology",
  "Multi-Objective Optimisation Ontology",
  "Probability & Statistics Ontologies",
  "Graph Theory Ontologies",
  "Markov Process Ontologies",
  "Game Theory Ontologies",
  "Decision Theory Ontologies",
  "Simulation Ontologies",
  "Mathematical Modelling",
  "Scientific Computing Ontologies"
 ]

 ROUTE_RULES = [
  "Operations Research -> Applied Mathematics Ontologies for parent semantics",
  "Operations Research -> Optimisation for mathematical programming",
  "Operations Research -> Probability & Statistics for uncertainty models",
  "Operations Research -> Graph Theory for network problems",
  "Operations Research -> Markov Processes for stochastic state systems",
  "Operations Research -> Game Theory for strategic interactions",
  "Operations Research -> Decision Theory for preference and utility reasoning",
  "Operations Research -> Simulation for complex system experimentation",
  "Operations Research -> Mathematical Modelling for system representation",
  "Operations Research -> Scientific Computing for large-scale computational execution"
 ]
}

UPDATE_MODEL {
 MODE = "append-compatible semantic versioning"

 MAJOR = "ontology-breaking conceptual change"
 MINOR = "new compatible OR model classes, algorithms or reasoning pathways"
 PATCH = "clarification, metadata or formatting correction"

 RULES = [
  "Preserve operational decision semantics.",
  "Preserve objective-and-constraint semantics.",
  "Preserve resource-allocation semantics.",
  "Preserve uncertainty-aware decision semantics.",
  "Preserve model-versus-real-system distinction.",
  "Preserve operational feasibility requirements.",
  "Preserve sensitivity and implementation reasoning.",
  "Preserve machine-readable operations-research semantics."
 ]
}

OUTPUT {
 CANONICAL_DEFINITION =
  "Operations research applies mathematical modelling, optimisation, probability and simulation to improve decisions and operational system performance."

 OPTIMISATION_SUMMARY =
  "Mathematical programming allocates resources and chooses decisions under explicit objectives and constraints."

 NETWORK_SUMMARY =
  "Network optimisation models routing, flow, assignment and location using graph structure."

 SCHEDULING_SUMMARY =
  "Scheduling coordinates tasks and limited resources over time."

 INVENTORY_SUMMARY =
  "Inventory models balance replenishment, holding, shortage and service-level costs."

 QUEUEING_SUMMARY =
  "Queueing theory analyses waiting, congestion, service capacity and utilisation."

 SIMULATION_SUMMARY =
  "Simulation evaluates complex operational systems when analytical solution is difficult or incomplete."

 DECISION_SUMMARY =
  "Decision analysis compares alternatives using outcomes, uncertainty, preferences and utility."

 IMPLEMENTATION_SUMMARY =
  "Operations-research solutions should be tested for sensitivity, robustness and operational feasibility before implementation."
}

FLOW {
 INPUT
 -> detect_operations_research_context
 -> identify operational_system_decisions_objectives_resources_constraints_uncertainty_stakeholders_and_performance measures
 -> classify mathematical_programming_network_scheduling_inventory_queueing_simulation_decision_analysis_stochastic_or_hybrid problem
 -> construct mathematical_or_computational model
 -> select linear_integer_nonlinear_dynamic_network_queueing_inventory_simulation_or_decision method
 -> solve_or_simulate system
 -> evaluate objective_values_service_levels_waiting_cost_risk_resource use_and_feasibility
 -> perform sensitivity_scenario_and_robustness analysis
 -> translate model_solution_into operational recommendation
 -> route to applied_mathematics optimisation probability_statistics graph_theory Markov_process game_theory decision_theory simulation mathematical_modelling_or_scientific_computing ontology
 -> recursively_expand_required_dependencies
 -> verify_result
 -> assign_confidence
 -> produce_semantic_output
 -> END
}

SEMANTIC_SUMMARY {
 "The Operations Research layer defines mathematical decision science for complex operational systems. It supports mathematical programming, resource allocation, linear and integer programming, network optimisation, transportation and assignment, facility location, vehicle routing, scheduling, inventory control, queueing theory, Markov models, simulation, decision analysis, utility, multi-criteria decision methods, game theory, reliability, risk, stochastic and robust optimisation and sensitivity-aware operational implementation."
}

FLOW_HASH {
 METHOD = "semantic-content-identity"
 ID = "KIRETH-MATH-APPLIED-MATHEMATICS-OPERATIONS-RESEARCH-V1"
}

}

END_FLOW